The most significant references have already been mentioned, like Romeo's quote of "this day's black fate" and then later ascribing himself as "Fortune's fool," but the whole play is wrapped around Shakespeare's notion of "star cross'd lovers," so from the very beginning, fate and the idea of destiny surrounds Romeo and Juliet's relationship. Shakespeare coined the word "disaster," which literally taken means an event that goes against the stars, much like Romeo and Juliet's love affair.
In Act III, Scene 1 of Romeo and Juliet, after Romeo enters and causes Mercutio to be mortally wounded, Benvolio, having just re-entered the scene, informs Romeo that Mercutio has just died. Romeo exclaims,
This day's black fate on more days doth depend;
This but begins the woe others must end. (3.1.120-121)
Romeo realizes that his act will cause grief and difficulty for Juliet, who is now his wife. But, he is determined to "end" the woe by avenging Mercutio's death.
Fate is a big player throughout the entire play. We are told in the prologue that fate is going to drive the story. In Act 3, fate plays with everyone. Mercutio dies because Romeo is fated for trouble. Think about it: If Tybalt wasn't after Romeo, and Mercutio wasn't friends with Romeo, Mercutio would not have died (not to mention Romeo trying to "help" and causing Tyablt to hit Mercutio). Juliet is affected by this duel, because Romeo is banished and her cousin dies. It also causes her parents to speed up her marriage to Paris, thus causing her parents to lose Juliet for good.
After slaying Tybalt, Romeo references Fate in saying (fortune is interchangeable with Fate):
O, I am fortune's fool!
He is speaking specifically of the painful irony of the situation in which he, Juliet's husband, accidentally causes Mercutio to be slain then slays Tyblat, when all he desired was to quell the feud with allusions to familial love (he is the only one who know their families are now joined). [Hint: Benvolio makes a reference to Fate a few lines after this.]
While Romeo might be the only one who specifically mentions fate, he is not the only one who seems to be in fate's hands. We see a lot of foreshadowing in this particular scene. Mercutio and Benvolio are discussing a seemingly unrelated topic but we can see the foreshadowing of the disastrous events to come. For instance, Mercutio says
Nay, an there were two such, we should have none
shortly, for one would kill the other.
We know that Mercutio is about to quarrel and be killed by Tybalt. There are many other examples in this section. Fate seems to take a cruel twist in this scene and the deaths of Mercutio and Tybalt serve as a catalyst for further cruel twists of fate in the rest of the play.
Romeo is the only character in the scene that specifically mentions fate. After Tybalt kills Mercutio, Romeo says:
This day's black fate on more days doth depend; This but begins the woe others must end.
After killing Tybalt, he again ascribes the events of the day to fate, claiming that he is "Fortune's fool" for having wound up in this situation.